Key needs to be in the ON/RUN position in order for ignition and fuel delivery to be active once RPM comes up, be it by the starter or pushing it. Even pre-fuel injection requires the key on to complete the ignition circuit and allow spark.
When the key is ON/RUN, for all intents and purposes, as far as the computer is concerned, the engine is "running", albeit 0 RPM. All that need happen is the engine be turned over in some manner and an RPM signal be produced for spark and fuel to be introduced and keep it self sustaining. All the start position does is additionally complete the circuit to the starter. The way it's set up in most cars, the computer is completely oblivious to the starter and can't tell the difference between the starter, a push start, a hand crank, letting the clutch out while rolling, whatever. It just sees an RPM and starts doing it's thing, and that means it has to be ON in order for that to occur. And "on" means anything from key in the ignition and in the ON/RUN position, including anti-theft transponder active and the proper RFID tag present and in range, etc.